×

zbMATH — the first resource for mathematics

Rotational transport on a sphere: local node refinement with radial basis functions. (English) Zbl 1303.76128
Summary: This paper develops an algorithm for radial basis function (RBF) local node refinement and implements it for vortex roll-up and transport on a sphere. A heuristic based on an electrostatic repulsion type principle is used to re-distribute the nodes, clustering in areas where higher resolution is needed. It is then important to have a scheme that varies the shape of the RBFs over the domain so as to counteract the effects of Runge phenomena where the nodes are sparse. The roll-up of two diametrically opposed moving vortices are studied. The performance differences between near-uniform and refined nodes are addressed in terms of convergence, time stability, and computational cost. RBF results are put into context by comparison with published results for methods such as finite volume and discontinuous Galerkin.

MSC:
76U05 General theory of rotating fluids
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bozzini, M.; Lenarduzzi, L.; Rossini, M.; Schaback, R., Interpolation by basis functions of different scales and shapes, Calcolo, 41, 77-87, (2004) · Zbl 1168.65315
[2] Buhmann, M.D., Radial basis functions: theory and implementations, Cambridge monographs on applied and computational mathematics, vol. 12, (2003), Cambridge University Press Cambridge
[3] Wyatt, R.E.; Trahan, C.J., Radial basis function interpolation in the quantum trajectory method: optimization of the multi-quadric shape parameter, Jcp, 185, 27-49, (2003) · Zbl 1040.81515
[4] Cheney, E.W.; Light, W.A., A course in approximation theory, (2000), Brooks/Cole New York · Zbl 0575.41001
[5] Cherkassky, V.; Mulier, F., Learning from data: concepts, theory, and methods, (1998), John Wiley New York · Zbl 0960.62002
[6] Driscoll, T.A.; Heryundono, A., Adaptive residual subsampling methods for radial basis function interpolation and collocation problems, Comput. math. appl., 53, 927-939, (2007) · Zbl 1125.41005
[7] Fasshauer, G.E.; Zhang, J.G., On choosing “optimal” shape parameters for RBF approximation, Numer. algorithms, 45, 345-368, (2007) · Zbl 1127.65009
[8] Flyer, N.; Wright, G.B., Transport schemes on a sphere using radial basis functions, J. comput. phys., 226, 1059-1084, (2007) · Zbl 1124.65097
[9] Flyer, N.; Wright, G.B., A simple radial basis function method for the shallow water equations on a sphere, P. roy. soc. A-math. phy., 465, 2106, 1949-1976, (2009) · Zbl 1186.76664
[10] Fornberg, B.; Flyer, N.; Hovde, S.; Piret, C., Locality properties of radial basis function expansion coefficients for equispaced interpolation, IMA J. numer. anal., 28, 1, 121-142, (2008) · Zbl 1134.65013
[11] Fornberg, B.; Piret, C., A stable algorithm for flat radial basis functions on a sphere, SIAM J. sci. comput., 200, 178-192, (2007)
[12] Fornberg, B.; Piret, C., On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere, J. comput. phys., 227, 2758-2780, (2008) · Zbl 1135.65039
[13] Fornberg, B.; Wright, G., Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. math. appl., 48, 853-867, (2004) · Zbl 1072.41001
[14] Fornberg, B.; Zuev, J., The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. math. appl., 54, 379-398, (2007) · Zbl 1128.41001
[15] Hubbert, S.; Morton, T., \(l^p\)-error estimates for radial basis function interpolation on the sphere, J. approx. theory, 129, 58-77, (2004) · Zbl 1065.41004
[16] Iske, A., Multiresolution methods in scattered data modelling, Lecture notes in computational science and engineering, vol. 37, (2004), Springer-Verlag Heidelberg
[17] Jetter, Kurt; Stöckler, Joachim; Ward, Joseph D., Error estimates for scattered data interpolation on spheres, Math. comput., 68, 226, 733-747, (1999) · Zbl 1042.41003
[18] Kansa, E.J.; Carlson, R., Improved accuracy of multiquadric interpolation using variable shape parameters, Comput. math. appl., 24, 99-120, (1992) · Zbl 0765.65008
[19] Madych, W.R.; Nelson, S.A., Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. approx. theory, 70, 94-114, (1992) · Zbl 0764.41003
[20] Micchelli, C.A., Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. approx., 2, 11-22, (1986) · Zbl 0625.41005
[21] Nair, R.; Jablonowski, C., Moving vortices on the sphere: a test case for horizontal advection problems, Mon. weather rev., 136, 699-711, (2008)
[22] Nair, R.D.; Thomas, S.J.; Loft, R.D., A discontinuous Galerkin transport scheme on the cubed-sphere, Mon. weather rev., 133, 814828, (2005)
[23] Powell, M.J.D., The theory of radial basis function approximation in 1990, (), 105-210 · Zbl 0787.65005
[24] M.J.D. Powell, Radial basis function methods for interpolation to functions of many variables, DAMTP Report NA11, University of Cambridge, 2001.
[25] Putman, W.M.; Lin, S.-J., Finite-volume transport on various cubed-sphere grids, J. comput. phys., 227, 1, 55-78, (2007) · Zbl 1126.76038
[26] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. comput. math., 11, 193-210, (1999) · Zbl 0943.65017
[27] Sarra, S.A.; Sturgill, D., A random variable shape parameter strategy for radial basis function approximation methods, Eng. anal. bound. elem., 33, 1239-1245, (2009) · Zbl 1244.65192
[28] Schoenberg, I.J., Metric spaces and completely monotone functions, Ann. math., 39, 811-841, (1938) · Zbl 0019.41503
[29] Schoenberg, I.J., Positive definite functions on spheres, Duke math. J., 9, 96-108, (1942) · Zbl 0063.06808
[30] Anton Sherwood, How can I arrange N points evenly on a sphere? Website, 2007, <http://www.ogre.nu/sphere.htm>.
[31] Wendland, H., Scattered data approximation, Cambridge monographs on applied and computational mathematics, vol. 17, (2005), Cambridge University Press Cambridge
[32] Wertz, J.; Kansa, E.J.; Ling, L., The role of multiquadric shape parameters in solving elliptic partial differential equations, Comput. math. appl., 51, 1335-1348, (2006) · Zbl 1146.65078
[33] Womersley, R.S.; Sloan, I.H., How good can polynomial interpolation on the sphere be?, Adv. comput. math., 23, 195-226, (2001) · Zbl 0980.41003
[34] R.S. Womersley, I.H. Sloan, Interpolation and cubature on the sphere, Website, 2003, <http://web.maths.unsw.edu.au/rsw/Sphere/>.
[35] Yoon, J., Spectral approximation orders of radial basis function interpolation on the Sobolev space, SIAM J. math. anal., 23, 4, 946-958, (2001) · Zbl 0996.41002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.